IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 12, DECEMBER Frequency-Domain Analysis of Linear Time-Periodic Systems

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1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 12, DECEMBER Frequency-Domain Analysis of Linear Time-Periodic Systems Henrik Sberg, Member, IEEE, Erik Möllerstedt, Bernhardsson Abstract In this paper, we study convergence of truncated representations of the frequency-response operator of a linear time-periodic system The frequency-response operator is frequently called the harmonic transfer function We introduce the concepts of input, output, skew roll-off These concepts are related to the decay rates of elements in the harmonic transfer function A system with high input output roll-off may be well approximated by a low-dimensional matrix function A system with high skew roll-off may be represented by an operator with only few diagonals Furthermore, the roll-off rates are shown to be determined by certain properties of Taylor Fourier expansions of the periodic systems Finally, we clarify the connections between the different methods for computing the harmonic transfer function that are suggested in the literature Index Terms Convergence analysis, frequency-response operators, linear time-periodic systems, series expansions Notation: Signals in continuous time on an interval where,, is finite belongs to When the interval is clear from the context, it will be left out in the notation Square-summable sequences belong to, the norm is finite The set of times continuously differentiable real functions in some open set is denoted by For, we denote by the -th partial derivative of with respect to its first second argument, respectively denotes the differentiation operator, integration denotes the real numbers, the integers, the set of complex numbers is the imaginary unit, is the imaginary axis, denotes complex conjugate of, is the adjoint of defined on sig- I INTRODUCTION IN THIS paper, we study linear operators nals in, where, Manuscript received November 10, 2004; revised June 2, 2005 Recommended by Associate Editor U Jonsson This work was supported by the Swedish Research Council under Project , by the Swedish Foundation for Strategic Research under Project CPDC H Sberg was with the Department of Automatic Control, Lund University, Sweden He is now with the California Institute of Technology, Control Dynamical Systems, Pasadena, CA USA ( henriks@cdscaltechedu) E Möllerstedt is with Combra Syd AB, SE Lund, Sweden B Bernhardsson is with the Ericsson Mobile Platforms AB, SE Lund, Sweden ( bob@controllthse) Digital Object Identifier /TAC We restrict ourselves to the set of bounded operators operators with finite induced norm, ie, We assume in the following that the given operator is bounded has a time-domain representation with a causal impulse response for all where belong to Often we identity the impulse response with the operator Conditions for representability of an operator as an integral equation (2) are given in, for instance, [1] It is well known that systems with finite-dimensional state-space realizations as well as infinite-dimensional models such as time-delay systems may be written on the form (2) We will often make the assumption that the impulse response has uniform exponential decay This means that there are positive constants such that for all In particular, this assumption implies boundedness of, since, for all This may be shown by using [2, Th IV7222] If there is a real positive number such that (1) (2) for all (3) then the operator (or the impulse response) is said to be periodic with period The impulse response of a time-invariant system satisfies for all A system with a finite-dimensional state-space realization with no direct term can be written as The impulse response of the system is given by,, where is the transition (4) /$ IEEE

2 1972 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 12, DECEMBER 2005 matrix for, see [3] If the matrices,, are -periodic, the impulse response satisfies (3) For simplicity, we do not deal with systems with direct terms in this paper Direct terms may be included in the analysis using techniques similar to those in [4] In the following, are scalar signals That is, we treat single-input single-output systems However, this is just for notational convenience Everything can be done for multiple-input multiple-output systems with only minor modifications It is well known, see, for example, [4], [5], that frequencydomain representations of periodic systems are infinite-dimensional operators The main goal of this paper is to develop intuition for the decay rate of the elements in these operators, to review some known results in the area A Previous Work The study of periodic systems has a long history in applied mathematics control One reason for the interest in periodic systems is that natural man-made systems often have the periodicity property (3) Some examples are oscillators used in communication systems, planets satellites in orbit, rotors of wind mills helicopters, sampled-data systems, ac power systems There is an excellent survey of periodic systems control in [6] Frequency-domain analysis of linear time-periodic systems in continuous time has been studied by several authors in the past A classical reference is the work by Zadeh, see [7], where the steady-state response of a time-varying system to harmonics is used to define a time-varying transfer function, the parametric transfer function (PTF) The PTF has been further developed used in [8], [9] The PTF is a scalar function that depends on two variables: time frequency A second frequency-domain representation can be obtained by using time-domain lifting on the time-periodic system, then applying the -transform, see [5], [10], [11] Colaneri called this function the transfer function operator (TFO) This is an integral operator with a kernel depending on time frequency An alternative derivation of the TFO comes from studies of the steady-state response of the system to an exponentially modulated periodic (EMP) signal The EMP signals serve as good test functions, since EMP signals are mapped to EMP signals by periodic systems A third approach was taken by Wereley Hall in [5], [12] They applied an harmonic balance method to state-space systems with EMP inputs That is, periodic matrices signals are exped into Fourier series harmonics are equated This method yields a transfer function that depends only on frequency The function was called the harmonic transfer function (HTF) The HTF is an infinite-dimensional operator The infinite dimensionality can be seen as the price that is paid for the removal of the time dependence in the transfer function For example, it was shown in [13] that a Fourier expansion of the PTF in the time direction yields the elements of the HTF All of the above transfer functions can be used for studies of periodic systems, for example, to compute norms It is important to underst that all of these transfer functions are equivalent The PTF the TFO are time dependent If this time dependence is expressed in an harmonic basis of the type, we essentially obtain the HTF Relations of this sort are treated in [13] [15] In this paper, we investigate what happens when higher harmonics in this basis are truncated An alternative approximation method is, for example, to use fast sampling in time Such approximations are discussed in [16] It should also be mentioned in this context that the frequency-domain operator of a discrete-time periodic system becomes finite dimensional, see [17], [18] The previous listing of frequency-domain methods is not complete There are more representations, see, for example, [19], [20] In the area of sampled-data systems, a lot of related work has been done The PTF has been applied to sampled-data systems in [8] A method similar to the TFO, that is, a lifting -transform approach, has been used in, for example, [15] [21] An approach similar to the HTF has been used in [22] [23] A nice property of sampled-data systems is that often closed-form solutions are obtained This is not the case for generic periodic systems The literature on sampled-data systems is vast many more references can be found in the previous work We mainly work with the HTF in this paper From the above discussion, it follows that this is not a severe restriction The HTF has successfully been used by several authors for different applications in the past For example, for identification of helicopter dynamics, see [18], for vibration damping in helicopters, see [24], for stability robustness analysis in switched power systems, see [25], [26] A nice feature with the HTF is that we can directly extract Bode-type diagrams that describe the cross-coupling of frequencies from the diagonals of the HTF This can be used to detect resonances that involve several frequencies, see [26] Another reason for studies of the HTF is that it has recently obtained a lot of theoretical attention Formally, we can work with the HTF just as with a stard transfer function Hence, formulas for -norms are completely analogous to the time-invariant formulas, see [4], [5], [27] However, the HTF is also useful for studies of attainable performance, see [28], generalization of the Nyquist criterion [29], for generalization of Bode s sensitivity integral [30], [31] B Computation of the HTF Despite all of this work, there are still open issues about the HTF In particular, how the HTF should be computed Three approaches have been taken, to the authors knowledge In the first approach, see [4] [5] it is assumed that has a state-space realization (4), that a Floquet transformation has been performed Then the matrix is time invariant, explicit formulas for the elements in the HTF can be given as a series of the Fourier coefficients of The second approach is also a state space approach, see [28] The elements of the HTF are given implicitly via an inversion of an unbounded quasi-toeplitz operator, with the Fourier coefficients of the state matrix on the diagonals It has been claimed that this yields the HTF when the dimension of finite-dimensional truncations of the operator grows toward infinity However, to the best knowledge of the authors of this paper, how when this convergence works has not been properly explained This second approach is interesting since it does

3 SANDBERG et al: FREQUENCY-DOMAIN ANALYSIS OF LINEAR TIME-PERIODIC SYSTEMS 1973 not require a Floquet transformation on the state-space model, allows us to work directly with the Fourier coefficients of the state-space realization We call this method the truncated harmonic balance method A third approach is used in [26] This approach is based on an impulse-response model (2) of the periodic system It is shown how the elements of the HTF can be computed via a Fourier expansion of the impulse response This approach is interesting since it only uses input-output data of the model However, the calculations in [26] are formal many details possibilities are not treated In Section VI of this paper, we clarify the connections between the above approaches, using the tools that are developed here In the limit, we obtain the same operator no matter what approach that is used Hence, it is justified to use the term HTF in all of the previous cases C Organization Contributions This paper serves the purpose of survey some existing results as well as introducing new results The new results give intuition for the structure of the HTF, based on two types of series expansions of the systems In Section II, we derive Taylor expansions of time-varying systems The expansions are around infinite frequency, the coefficients become time-varying Markov parameters Two different expansions are studied We introduce the concepts of input output roll-off of a time-varying system, relate the Markov parameters to the roll-off concepts In Section III, we derive Fourier expansions of time-periodic systems The generalized Fourier coefficients become time-invariant systems Similar ideas were suggested in, for example, [7], [26] Here we apply Hilbert space formalism to the problem Furthermore, we derive conditions under which truncated Fourier series converge in induced -norm, introduce the concept of skew roll-off In Section IV, we define the HTF based on the impulse response, as was done in [26] Its definition is straightforward after a Fourier expansion We show that if the periodic system has high input output roll-off, then it may be well approximated by a finite low-dimensional matrix function If the system has high skew roll-off, then it may be approximated by an operator with only few diagonals In Section V, we obtain error bounds for the closed-loop operator, when it is computed from truncated HTFs These formulas are useful in Section VI, where we show that the HTF defined in Section IV is identical to the HTF defined in [4], [5] We also study the truncated harmonic balance method It is seen that by a minor modification of the method, we can show that it converges to the desired operator The convergence may, however, be quite slow An early version of this paper is [32] see, for example, [33], A Markov Parameters for Time-Varying Systems As a first step in the analysis, we make an expansion of the convolution integral (2) that resembles a Taylor expansion This is motivated by the Markov parameters for time-invariant systems That is, a transfer function of a time-invariant system, where is the impulse response, can under certain regularity conditions be Taylor exped as as (5) The Markov parameters are they determine the response to high-frequency signals If the first Markov parameters are zero, then high-frequency signals are attenuated quickly ( The system has high roll-off ) We will establish similar conditions in the time-varying case start by putting restrictions on the impulse response, for the computations to be justified The set of continuously differentiable exponentially bounded impulse responses will appear frequently throughout this article Definition 1 (The Set ): A causal time-varying (not necessarily periodic) real impulse response belongs to the set if E1) belongs to, where ; E2) all its partial derivatives up to order have a limit everywhere on the boundary of, that is exists for all, where or, ; E3) all its partial derivatives up to order have uniform exponential decay Example 1 (State Space Models): We can check that the impulse response of state-space models belong to if the model is exponentially stable, belong to, belongs to Furthermore, all the matrices should be bounded over It will be useful to consider signals in the space of Schwartz functions, is bounded for all where is the differentiation operator The set is dense in, for, the Fourier transform of an element in is again in, see [34] To obtain expansions of (2) in the form of (5), we proceed by using integration by parts If we choose an input signal, this gives II TAYLOR EXPANSIONS OF TIME-VARYING SYSTEMS We will obtain a frequency-domain description of Many times the input signal output signal in are represented by their Fourier transforms, where is the angular frequency This presents no problems since is isomorphic to under the Fourier transform, (6)

4 1974 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 12, DECEMBER 2005 is the integration operator:, means the partial derivative of with respect to its second argument The above computation is allowed if One should notice that (6) is an expansion of the time-varying system (2) into a sum of a modulated integrator a new stable time-varying system with as impulse response After the following definition, we obtain the general expansion formulas Definition 2 (Input Output Markov Parameters): For a system with impulse response in, the input Markov parameters,, are defined as, the output Markov parameters, (7), are defined as then to integrate over in the -direction (11) Repeat this procedure on the virtual output for substitute into (11) The above computations are allowed under the given assumptions, since the outputs are continuously differentiable, as Equation (10) may also be proven from (9) by a duality argument, see Remark 2 Remark 2 (Duality of Input Output Markov Parameters): The adjoint of, where, isgiven by (the anti-causal relation), one can make Taylor expansions of this relation as well (8) for Remark 1 (Markov Parameters for Time-Invariant Systems): For time-varying systems in, the Markov parameters are bounded continuously time-varying functions For time-invariant systems, with impulse response, the input output Markov parameters coincide with the traditional Markov parameters, are constant equal Theorem 1 (Taylor Expansions of Time-Varying Systems): Assume that belongs to Then for every input, the output given by (2) can be expressed in either of the following two ways: Input Markov parameter expansion where So with the interchange the input Markov parameters of are the output Markov parameters of (with the obvious changes from causality to anticausality) B Input Roll-Off Output Roll-Off Equation (5) shows that there is a relation between the Markov parameters high-frequency behavior for time-invariant systems This relation will be further explored for time-varying systems We need the projection operator on that is defined by Output Markov parameter expansion (9) (10) Notice that is not causal in the time domain, It is also convenient to define This term is motivated in Section IV-A Definition 3 (Rectangular Truncation): Assume that Then is called a rectangular truncation of The systems in the following are strictly proper (they have no direct term) can be arbitrarily well approximated by its low-frequency part: To quantify the rate of convergence, we use the bound as Proof: To prove (9), use integrations by parts on, substitute into (6) If the procedure is repeated, we obtain (9) The first step in proving (10) is to differentiate (2) Definition 4 (Input Output Roll-Off): Assume that is a bounded operator on If there are positive constants such that

5 SANDBERG et al: FREQUENCY-DOMAIN ANALYSIS OF LINEAR TIME-PERIODIC SYSTEMS 1975 then is said to have output roll-off The largest such is called the maximum output roll-off If there are positive constants such that TABLE I INPUT AND OUTPUT MARKOV PARAMETERS OF A TIME-VARYING STATE SPACE MODEL (4) then is said to have input roll-off The largest such is called the maximum input roll-off Remark 3: The roll-off rates are not necessarily integers, but sometimes the maximum roll-off rates are, see Theorem 2 If a system has output roll-off, then it also has output roll-off, where, similarly for input roll-off Some simple properties for calculations with systems with input/output roll-off are stated in the following proposition Proposition 1 (Input Output Roll-Off): The following rules apply to systems with roll-off i) If has output roll-off is bounded, then has output roll-off of If has input roll-off is bounded, then has input roll-off of ii) Input output roll-off reduce to the stard notion of roll-off for time-invariant That is,, where is either the input or the output roll-off of iii) If is a time-invariant system with output roll-off has output roll-off, then has output roll-off If is a time-invariant system with input roll-off has input roll-off, then has input roll-off Proof: In this proof, means the induced -norm i) follows from Definition 4 the induced norm property ii) commutes with a time-invariant We have, where could be either the input or output roll-off iii) We have, which proves the first statement The second statement follows from a dual argument Since Definition 4 may be hard to check for a given operator, it simplifies if we decompose the system into terms that are easier to analyze The Taylor expansions in Theorem 1 are such decompositions We use them in the following theorem to show that the Markov parameters in Definition 2 determine the roll-off Theorem 2 (Markov Parameters Roll-Off): Assume that the impulse response of belongs to Let be the first nonzero input Markov parameter be the first nonzero output Markov parameter Then i) has input roll-off If for all, then has maximum input roll-off ; ii) has output roll-off If for all, then has maximum output roll-off Proof: We start with the second statement of i) We need a bound on Bydefinition, we have (12) To bound (12), we make an input Markov parameter expansion of, for Since is dense in, such a bound holds for inputs in as well Using the assumption that the first Markov parameters are zero,wehave The first term in the expansion can be bounded as where make the upper bound For the remaining terms, we can where, where Hence, using the triangular inequality we have (13) the system has input roll-off To see that is the maximum input roll-off, assume has a larger roll-off Then, we obtain a contradiction for sufficiently large in (13) due to the lower bound To prove the first statement of i), we make an input Markov parameter expansion as before, but we stop the expansion when we have terms with Then we can prove an upper bound similar to (13) (but not a lower bound) the result follows To prove ii), the output Markov expansion is used instead of the input Markov expansion Example 2 (Finite-Dimensional State Space Models): Let us assume that the system has a state space realization (4) that the impulse response belongs to, see Example 1 According to Theorem 2 the roll-off of the state space system can be determined by checking which of the Markov parameters are zero The first few Markov parameters of (4) are given in Table I In particular, they coincide with the normal Markov parameters for time-invariant systems: The Markov parameter conditions for output roll-off correspond to the conditions for relative degree of a periodic

6 1976 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 12, DECEMBER 2005 system, defined in [35] Also notice that the -th Markov parameters may be written as, where, using notation from [35] with convergence in The Fourier coefficients are given by (18) III FOURIER EXPANSIONS OF TIME-PERIODIC SYSTEMS Until now we have not used the periodicity condition The periodicity condition can be used for Fourier expansions The possibility of Fourier expansions of periodic systems was discussed already in [7] Here, we apply Hilbert space formalism to the problem We define the space for periodic impulse responses where is periodic causal (19) Proof: We show (16) A similar calculation gives (17) Using the Cauchy Schwartz inequality on (18), we have for all Since Fourier series, we can exp it in the generalized The previous equality follows from the periodicity condition (3) The -norm is also used in, for example, [4], [10], [28] is a Hilbert space with the scalar product (14) is composed of the well-known separable Hilbert spaces Orthonormal basis functions in are, for example or where is an orthonormal basis in We can choose, where are the Laguerre polynomials, for example By stard results from functional analysis, see, for example [36], all functions in can be represented by a generalized Fourier series If we insert this in the sum in (16), we see that it is equal to (15) Some immediate properties of the Fourier coefficients in (16), (17) are stated in the following corollary Corollary 1 (Properties of Fourier Coefficients): Assume that belong to Then i) the Fourier coefficients satisfy ; ii) if is real, then for all ; iii) the Fourier coefficients belong to for all ; iv) the scalar product (14) can be expressed as By Corollary 1 it follows that there is no essential difference between the expansions (16) (17) They are essentially the same In the following sections, we will mostly work with the expansions in It is also useful to introduce the orthogonal projection (15) where the series converge in -norm Instead of using the expansion (15), it is more useful for us to only use the expansion in the -direction This is expressed in the following theorem Theorem 3 (Fourier Expansion in ): Assume that the impulse response of belongs to Then (16) (17) (20) We make the following definition Definition 5 (Skew Truncation): Assume that the impulse response of is in Then the system, with impulse response given by (20), is called an th-order skew truncation of The reason for the term skew truncation will be more clear once we have constructed the harmonic transfer function in Section IV-B A simple application of Corollary 1 iv) gives that

7 SANDBERG et al: FREQUENCY-DOMAIN ANALYSIS OF LINEAR TIME-PERIODIC SYSTEMS 1977 the skew truncations converge in,,as Remark 4 (Optimal Approximations in ): By Corollary 1 iv), impulse responses that do not contain the same Fourier coefficients are orthogonal in Hence, in From stard results for approximation in Hilbert spaces, is the optimal -approximation of in the subspace In particular, is the optimal time-invariant impulse-response approximation of in A Input Output Properties The interpretation of the convergence in the above Fourier expansions is that for impulse inputs, the outputs converge in mean energy sense This is a quite weak form of convergence By strengthening the assumptions on, we can show stronger forms of convergence, in induced norms This is the topic of the rest of this section The input output map of is given by (21) where we have interchanged the order of integration summation The output is given by a parallel connection of input- or output-modulated time-invariant systems We associate with the th Fourier coefficients of causal time-invariant system Proof: i) We will prove that belongs to Since belongs to at least, by E3) we have Hence, we can bound the -norm ii) By assumption E1) we have that for all Make a Fourier expansion in the -direction of, notice that Hence, we have the bound (23) for some positive constants Such constants exist by assumption E3) The result follows Using Lemma 1 we can show the following theorem Theorem 4 (Convergence of Skew Truncations): Assume that a periodic system has an impulse response that belongs to, where Then, for all inputs, the output in (21) converges, uniformly in, to in (2), as Under the same assumptions on, we have the following convergence bound on the Fourier series (22) (24) for a system-dependent constant Proof: We start with the first statement Using Lemma 1 ii) we have that Hence, we can represent by the formal Fourier series (22) where the Fourier coefficients are time-invariant systems We will show that these series converge in induced norms Let us again use the class of causal continuously differentiable exponentially bounded impulse responses,,defined in Section II Lemma 1 (Bounded Fourier Coefficients): Assume that a periodic system has an impulse response that belongs to Then i) belongs to ; ii) there are positive constants such that the Fourier coefficients are bounded where is a constant such that (25) (26) Since the upper bound in (25) is independent of tends to zero as, the convergence is uniform Furthermore, we have shown (24) when, with It remains to prove (24) for We use that the Schwartz functions are in, are dense in, If the domain of is restricted to, we can by (25) interchange the order of integration summation represent

8 1978 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 12, DECEMBER 2005 by the Fourier series, where are time-invariant systems Furthermore, we have the bound using first Young s inequality then Lemma 1 ii) Now, since is dense in,, we can conclude that then has skew roll-off From Theorem 4 we see that has skew roll-off if belongs to Remark 5 (Skew Roll-Off of Time-Invariant Systems: For a time-invariant system, it holds that for all Hence, a time-invariant system has infinite skew roll-off IV HARMONIC TRANSFER FUNCTION (27) where is a given by (26) Hence, also in this case We can apply Theorem 4 to input signals that are harmonics thereby see the connection between the above Fourier expansions the classical analysis by Zadeh [7] Example 3 (Harmonic Response the PTF): The response of a periodic system to an harmonic is now easy to obtain Under the assumptions of Theorem 4 by the definition of the Fourier transform we obtain (28) where is the Fourier transform of Since, we know that is uniformly continuous belongs to, see [33] Equation (28) shows that the response includes a countable number of frequencies separated by multiples of The parametric transfer function (PTF) that is used in [7] [9] may be defined as the steady-state response of the periodic system to an harmonic, By the previous analysis, we realize that The PTF can also be computed directly from the impulse response, see [7] [9] The HTF that we obtain in Section IV, is also a frequency-domain representation of The difference is that the PTF is scalar but depends on time frequency, whereas the HTF only depends on frequency The price is that the HTF becomes infinite dimensional The relation between the PTF the HTF has also been discussed in [13] B Skew Roll-Off In analogy with input roll-off output roll-off we define skew roll-off Definition 6 (Skew Roll-Off): If there are positive constants such that Time-periodic systems can be lifted to formally time-invariant systems using various techniques, see, for example, [21], [37] In this section, we review one such representation, the harmonic transfer function (HTF) All lifted representations have one thing in common: they are infinite-dimensional operators Here we apply the Taylor the Fourier expansions of the previous sections to show how the HTF can be approximated It also turns out that the roll-off concepts have a clear interpretation for the HTF By including a sufficient amount of frequencies in the Fourier expansion of, we can come arbitrarily close to itself, as discussed in Section III Since we have decomposed the periodic system into time-invariant terms, the frequency-domain analysis is now straightforward Assume in the following that the assumptions of Theorem 4 hold Notice that from (2) may be written as (29) where is the stard convolution product Now pick an input in We can apply the Fourier transform on both sides of (29), get (30) All the Fourier transforms are well defined, since by the assumptions By Theorem 4, converges to in Furthermore, are isomorphic under the Fourier transform Hence, for all inputs, converges to in as Therefore we can put in (30) if we mean convergence in -sense, not point-wise convergence Next we rewrite the summation (30) by using lifting on In [22] this lifting was called the Sample-Data Fourier transform (SD-transform) The SD-transform is an isometric isomorphism between a Hilbert space we denote by It maps the Fourier transform into an infinite-dimensional column-vector-valued function The SD-transform of is denoted by is defined as Since the vector contains repeated versions of,itis enough to define for to be

9 SANDBERG et al: FREQUENCY-DOMAIN ANALYSIS OF LINEAR TIME-PERIODIC SYSTEMS 1979 able to take the inverse SD-transform We define the norm in as (33) For signals, there are now three representations:,,, the following extended Plancherel s theorem holds, If has finite -norm, then is in (its elements are square summable) for almost all, that is almost everywhere Using the SD-transform, (30) can be written in matrix-vector form as when if we put (31) A Rectangular Truncations Revisited By looking at the structure of the HTF s of rectangular skew truncated systems we make useful connections to the work in [27] The reasons for the terms rectangular skew will also be obvious To compute the induced -norm (1) of a system with input output roll-off,wehave the bound (34) Proposition 2 gives us a way to compute the induced -norm, given a HTF It is not essential that corresponds to a causal operator for (33) to hold, it is true for every frequencydomain relation (31) Hence, we can apply it to This is favorable, since the HTF of is simple Proposition 3 (Rectangular Truncated HTFs): Assume that has an HTF If we choose, introduce the intervals, the HTF of is given by where is the part of that maps frequencies in to frequencies in, that is We call the HTF of it can be regarded as a linear operator on for each HTF was the term used by Wereley in [5] A similar object was called the FR operator in [22] in the case of sampled-data systems The difference between these efforts is the way the elements of are computed In the sampled-data case, explicit formulas are given in [22] In the time-periodic state-space case formulas are given in [4], [5], in the impulse response case formulas are given here in [26] The relation between the impulse-response state-space approaches is further discussed in Section VI It was assumed in the above discussion that the impulse response belongs to This was done to motivate the construction of the HTF from an input-output view However, the HTF is a meaningful construction as soon as its elements, the Fourier transforms of the Fourier coefficients of, are well defined This is the case, for example, when is in Wehave the following well-known results, which are derived in [4], [5], [22] under slightly different assumptions Proposition 2 (Norm Formulas): Assume that the impulse response of the periodic system belongs to Then can be defined as in (31) We have then trace (32) where Proposition 3 shows that we can represent the linear periodic system arbitrarily well with finite-dimensional functions estimate its norm as (35) using (33) assuming continuous elements The guaranteed accuracy depends upon the matrix size the roll-off of according to (34) To use the rectangular truncation (35) to estimate the norm of a periodic system has been suggested by many authors A similar idea is to use a compression operator, see [38] In [22], it is shown that the rectangular truncation converges at least at a rate, for truncations some constant In [22], sampled-data systems are studied, but similar techniques are used in [4], [27] As seen in (34), the bound on convergence rate may be improved by checking the Markov parameters Moreover, nonsquare truncations can be used to improve convergence computation time, adapting to the input output roll-off rates

10 1980 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 12, DECEMBER 2005 Input output roll-off can be visualized in the HTF as follows Introduce the interval look at the HTFs corresponding to the truncated operators in the definition of input output roll-off, Definition 2, 4, use notation from Proposition 3 the -norm is computed for the system, with impulse response Then (36) (37) where is the (Floquet-transformed) state matrix, see the discussion in Section I-B The HTF of has diagonals, but notice that in general Hence, are in general not -optimal approximations of, see Remark 4 V APPROXIMATE INVERSES for almost all from (33) Since the modulus of the elements in an operator on cannot be larger than the induced -norm, this also bounds the sizes of individual elements To conclude, if the system has high output roll-off, then decays quickly asymptotically in the up-down direction (36), if it has high input roll-off it decays quickly asymptotically in the left-right direction (37) In Theorem 2 we have given the conditions for the decay rates: the more input output Markov parameters that are zero, the higher roll-off B Skew Truncations Revisited Let us now look at the skew truncated HTF s Proposition 4 (Skew Truncated HTF s): The HTF of the skew truncation,, consists of the diagonals of is zero elsewhere Hence, as increases, more diagonals are added to the skew truncated HTF Furthermore, we know that each diagonal represents a Fourier coefficient of, see Theorem 3 For instance, the middle diagonal corresponds to the time-invariant component, see Remark 4 From Theorem 4 we know how quickly this HTF converges in induced norms That is, we can quantify how much accuracy there is, at least, to gain by including an extra diagonal in the HTF If the system has skew roll-off,we can conclude that Inverses appear for closed-loop systems in mappings such as Here, we study what can be said about the approximate inverses, using the machinery developed thus far This is of interest by itself, in studies of closed-loop systems, for instance, but in this paper we only use it for state space models in Section VI We should first remember that even if is causal, the approximation is noncausal, even if it gets less noncausal as it converges to For this reason it is difficult to prove causality of by studying However, to compute, we have the following result: Proposition 5 (Approximation of Using ): Assume that are bounded operators on, that has output roll-off input roll-off Then the relative -induced norm error is bounded Proof: All norms in this proof denote -induced norms First, we make an orthogonal decomposition of the Hilbert space, so that In this basis, takes the operator-matrix form where, using Definition 6 (33) Hence, with high skew roll-off, the diagonals decay quickly for large The constants can be determined from the smoothness of, see Theorem 4 Remark 6 (Relation to [27]): Skew truncations are used to compute the -norm of periodic systems in [27] However, it is not the HTF of that is used there Instead the state-space matrices are skew truncated That means that where,,, Next, we notice that

11 SANDBERG et al: FREQUENCY-DOMAIN ANALYSIS OF LINEAR TIME-PERIODIC SYSTEMS 1981 that second factor on the right-h side in matrix form becomes matrices To use the results of the previous sections, be interpreted as the (induced) Euclidean norm should A Floquet-Transformed State Space Models Assume that a Floquet transformation, see, for example [3], has been performed on the state space realization (4) of, that the Fourier series The norm of the first factor is bounded by, the second factor is bounded by, the result follows The approximation is a finite sum of modulated timeinvariant causal systems If we decompose as, we have bounds on from Theorem 4 Assume that is causal bounded Then, notice that are absolutely convergent The state matrix is constant, Then, the Fourier series of the impulse response is given by (38) If is large enough we have,, we can make the following Neumann series expansion with absolute convergence in induced -norm: after interchange of summation order Hence, the Fourier coefficients of, see Section III, are given by (39) If the approximation error is small, then boundedness of follows We formalize this in the following proposition Proposition 6 (Approximation of Using : Assume that is a bounded causal operator on, where, that where Then given by (38), (39) is a bounded causal operator on, the relative error is bounded is iden- Using the definition (31) of the HTF, we see that tical to the HTF defined in [4] [5] B Convergence of the Truncated Harmonic Balance Method As discussed in Section I-B, it is of interest to compute the HTF of a state space model without first applying the Floquet transform Truncated harmonic balance was suggested as a method for this in, for example, [28] However, to the authors knowledge, no analysis of how when this method converges has been presented We will do an attempt to analyze the method here Define the multiplication operator as (40) Proof: That is bounded causal follows since it is a product of the bounded causal operators in (38) The bound (40) follows from a simple bound on the geometric series (39) Proposition 6 is stronger than Proposition 5 in the sense that we do not need to assume existence of the approximate inverse, the existence follows by the Neumann series expansion On the other h, is easier to compute, since it can be represented by matrices using Proposition 3 VI STATE-SPACE MODELS Now, we return to the state space systems described in (4), show how the results in the previous sections can be applied to this situation Here it is useful to allow signals to be vectors or similarly The input output relation of the state space model (4) is then given by (41) for all, where is the differentiation operator The reason for introducing is to make all operators bounded Equation (41) is decomposed of three simple operators The operators have impulse responses of the type (42)

12 1982 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 12, DECEMBER 2005 the HTF becomes is time invariant, all the operators are diagonal If we approximate, we obtain where the second factor is a Toeplitz operator are the Fourier coefficients of It is now straightforward to check the input output Markov parameters If belong to, the first Markov parameters of are, respectively, hence both have input output roll-off 1 by Theorem 2 If both belong to, then have skew roll-off, by Theorem 4 Let us focus on the rectangular truncations, since these are suggested in [28] Under the assumptions of Proposition 5 using Proposition 3, we have bounds The first part of the error bound depends on the input output roll-off of, which are determined by the Markov parameters in Table I The second part depends on the operators,,, as discussed previously To summarize, using techniques developed in this paper, we have shown convergence of the truncated harmonic balance method suggested in [28] Notice that the method suggested here in [28] are the same, since commute We have also seen that the worst-case convergence rate is slow, only for matrix dimensions The advantage with the method is that we only work with the Fourier coefficients simple matrix algebra No knowledge of the transition matrix is needed If the transition matrix is known, we can compute the elements in exactly by our results in Section IV Then the convergence of rectangular truncations may be much faster, depends only on the input output roll-off Since the convergence rate of the rectangular-truncation method may be slow, it is an interesting problem for future research to study how the method may be improved VII CONCLUSION as, To approximate, we use skew truncations, since is a multiplication operator has no input output roll-off If is periodic in, we have that an th-order skew truncation converges as for some constant The HTF of is a Toeplitz operator with the Fourier coefficients on the diagonals, see [4] Hence, by rectangular truncations of each of the operators in (42), we can approximate with (43) as, for some constants,,, using the triangular inequality Notice that (43) is a worst-case bound If the system We have studied linear time-periodic systems from a frequency-domain point of view in this paper We started to study Taylor expansions of time-varying systems defined input output Markov parameters We also introduced the concepts of input output roll-off These roll-off rates are determined by the Markov parameters Next we studied Fourier expansions of periodic systems in We also gave sufficient conditions for convergence rates of truncated Fourier expansions in induced -norm, introduced the concept of skew roll-off After the Fourier expansion, it was straightforward to define the frequency-response operator that is called the HTF The roll-off concepts were shown to determine the decay rates of elements in different directions of the HTF, we were able to strengthen available convergence bounds After studies of inverses, we applied the results to systems given in state-space form This allowed us to give conditions under which the truncated harmonic balance method converges This method is interesting since only the Fourier coefficients of the realization are needed Most other methods that apply to periodic systems require knowledge of the transition matrix However, the convergence rate of the method can be quite slow This paper has provided a systematic convergence analysis for the HTF This is important since in all applications listed in Section I-A, some sort of truncation is used We have analyzed the most common approaches of truncation here However, it is still unclear how the HTF is best approximated

13 SANDBERG et al: FREQUENCY-DOMAIN ANALYSIS OF LINEAR TIME-PERIODIC SYSTEMS 1983 ACKNOWLEDGMENT The authors would like to thank J Malinen, A Rantzer, KJ Åström for fruitful discussions suggestions The first author spent the spring of 2003 at the Mittag Leffler Institute, Stockholm, Sweden, he is thankful to the institute its staff REFERENCES [1] I W Sberg, Integral representations for linear maps, IEEE Trans Circuits Syst, vol 35, no 5, pp , May 1988 [2] C A Desoer M Vidyasagar, Feedback Systems: Input-Output Properties New York: Academic, 1975 [3] W J Rugh, Linear System Theory Upper Saddle River, NJ: Prentice- Hall, 1996 [4] J Zhou T Hagiwara, Existence conditions properties of the frequency response operators of continuous-time periodic systems, SIAM J Control Optim, vol 40, no 6, pp , 2002 [5] N Wereley, Analysis control of linear periodically time varying systems, PhD dissertation, Dept Aero Astro, Mass Inst Technol, Cambridge, MA, 1991 [6] S Bittanti P Colaneri, Periodic 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McKean, Fourier Series Integrals New York: Academic, 1972 [34] L Hörmer, The Analysis of Linear Partial Differential Operators I: Distribution Theory Fourier Analysis, 2nd ed New York: Springer-Verlag, 1990 [35] G De Nicolao, G Ferrari-Trecate, S Pinzoni, Zeros of continuous-time linear periodic systems, Automatica, vol 34, no 12, pp , 1998 [36] E Kreyszig, Introductury Functional Analysis With Applications New York: Wiley, 1978 [37] P Voulgaris, M Dahleh, L Valavani, H H optimal controllers for periodic multirate systems, Automatica, vol 30, no 2, pp , 1994 [38] G Dullerud, Computing the L -induced norm of a compression operator, Syst Control Lett, vol 37, no 2, pp 87 91, 1999 Henrik Sberg (S 02 M 05) was born in Staffanstorp, Sweden, in 1976 He received the MS degree in engineering physics the PhD degree in automatic control, both from Lund University, Lund, Sweden, in , respectively Since September 2005, he has been a Postdoctoral Scholar with Control Dynamical Systems, the California Institute of Technology, Pasadena His research interests include modeling, model reduction, time-varying systems, linear systems Dr Sberg was the winner of the Best Student- Paper Award at the IEEE Conference on Decision Control in 2004 Erik Möllerstedt was born in Lund, Sweden, in 1968 He received the MS degree in engineering physics the PhD degree in automatic control, both from Lund University He is currently working at Combra, Lund, Sweden, as a Consultant in product development His research interests are in linear control theory, physical modeling, simulation Bo Bernhardsson was born in Malmö, Sweden, in 1963 He received the MS degree in electrical engineering the PhD degree in automatic control from Lund University, Lund, Sweden He became a Professor in Automatic Control in 1999 Since 2001, he has been with Ericsson, Lund, Sweden, as a Senior Specialist in control system design for mobile terminals, focusing on the deployment of the third generation universal mobile telephony system His research interests are in linear control theory, signal processing, control of systems involving communication networks